chapter 6 solution methods for algebraic equations
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Instructor Tao, Wen-Quan
Key Laboratory of Thermo-Fluid Science & EngineeringInt. Joint Research Laboratory of Thermal Science & Engineering
Xi’an Jiaotong UniversityInnovative Harbor of West China, Xian
2020-Oct-19
Numerical Heat Transfer (数值传热学)
Chapter 6 Solution Methods for Algebraic Equations
6.1 Introduction to Solution Methods of ABEqs
6.2 Construction of Iteration Methods of LinearAlgebraic Equations
6.3 Convergence Conditions and AccelerationMethods for Solving Linear ABEqs.
6.4 Block Correction Method –PromotingConservation Satisfaction
6.5 Multigrid Techniques –PromotingSimultaneous Attenuation of DifferentWave-length Components
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6.1.1 Matrix feature of multi-dimensionaldiscretized equation
6.1.2 Direct method and iteration methodfor solving ABEqs.
6.1 Introduction to Solution Methods of ABEqs
6.1.3 Major idea and key issues of iterationmethods
6.1.4 Criteria for terminating iteration
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6.1 Introduction to Solution Methods of ABEqs
6.1.1 Matrix feature of multi-dimensionaldiscretized equation
For 2-D, 3-D flow and heat transfer problems, the
discretized equations with 2nd order accuracy:
2-D P P E E W W N N S Sa a a a a b
3-D P P E E W W N N S S F F B Ba a a a a a a b
For a 2D case with L1XM1unknown variables, the
general algebraic equation of kth variable is:
1 1
1
, , ,2 2 , 1 1 , 1 1 1
,
1 , 1 1
1, 1 1 , 1 1 , 1 1
.... ...
... ...
k k k k L k L k k L k L k k k
Lk k k k k L k M L M kk k k L k
a a a a a
a a a ba
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For 2-D problem with 2nd order accuracy there are
only five coefficients at the left hand side are not equal
to zero, and the matrix is of quasi (准)five-diagonal, a
large scale sparse matrix (大型稀疏矩阵).
If the 1-D storage
of the coefficients is
conducted as shown
right，then the order
of coefficients in one
line are:
, , , ,S W P E Na a a a a
PaWaEa Na
1 1
1
, , ,2 2 , 1 1 , 1 1 1
,
1 , 1 1
1, 1 1 , 1 1 , 1 1
.... ...
... ...
k k k k L k L k k L k L k k k
Lk k k k k L k M L M kk k k L k
a a a a a
a a a ba
Sa
0 0 0
0
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Features of ABEqs. of discretized multi-dimensional
flow and heat transfer problems:
1) For conduction of const. properties in uniform grid—
matrix is symmetric and positive definite(正定、对称）;2) For other cases: matrix is neither symmetric nor
positive definite.
ABEqs. of large scale sparse matrix （大型稀疏矩阵）are usually solved by iteration methods.
6.1.2 Direct method and iterative method forsolving ABEqs.
1.Direct method(直接法)
Accurate solution can be obtained via a finite timesof operations if there is no round-off error, such asTDMA，PDMA.
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From an initial field the solution is progressively
improved via the ABEqs. and terminated when a pre-
specified criterion is satisfied.
2. Iterative method（迭代法)
The ABEqs. of fluid flow and heat transfer problems
usually are solved by iteration methods :
1)Non-lineairity of the problems，the coefficients need
to be updated. There is no need to get the true solution
for temporary （临时的）coefficients;
2) The operation times of direct method is proportional
to N2.5~3，where N is the number of unknown variables.
When N is very large the operation times becomes
very very large, often unmanageable! 8/53
1. Major idea
In matrix form the ABEqs. is ： .A b 1( )A b
in multi-dimensional space R (the number of
( )1
k
A b （ ）k when( ) ( 1)
( , , )k k
f A b
2. Key issues of iteration methods
2) Is the series converged？
6.1.3 Major Idea and Key Issues of Iteration Methods
Its solution is
. Iteration method is to construct a series of
k
For the kth iteration
dimensions equals the number of unknowns) such that
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1) How to construct the iteration series of ？k
3) How to accelerate the convergence speed？
6.1.4 Criteria for terminating (inner) iteration(1) Specifying iteration times；
(2) Specifying the norm of p’eq.
residual less than a certain
small value；
(3) Specifying the relative norm
of p’eq. residual less than a
certain small value；
(4) Specifying relative change
of variable less than a small
value；
( 1)
( 1) ( )
max max
;k
k k
( 1) ( )
( 1)
0 max
k k
k
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6.2.1 Point (explicit) iteration
6.2.2 Block (implicit) iteration
6.2 Construction of Iteration series of
for solving Linear Algebraic Equations
6.2.3 Alternative direction iteration－ADI
k
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6.2 Construction of Iteration Methods of LinearAlgebraic Equations.
6.2.1 Point (explicit) iteration
The updating (更新) is conducted from node to node;
After every node has been visited a cycle (轮) of
iteration is finished; The updated value at each node is
explicitly related to the others.
In the updating of every node the previous cycle
values of neighboring nodes are used; The convergence
speed is independent of iteration direction.
1. Jakob iteration
2. Gauss－Seidel iteration
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3. SOR/SUR iteration
( 1)( 1) ( ) ( )( )kk k k
1 Under-
1 Over-(0 2)
Remarks：This relaxation is for solving the linear ABEqs.,
Not for the non-linearity.
6.2.2 Block (implicit) iteration （块隐式）
1. Basic idea
Dividing the solution domain into several regions,
within each region direct solution method is used, while
from block to block iteration is used，also called implicit
iteration.
Present values are used for updating.
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2. Line iteration（线迭代)-the most fundamental
of block iteration
The smallest block is a line: At the same line TDMA
is used for direct solution, from line to line iterative method
is used.
Solving in N-S direction and scanning (扫描) in E-W D.：
( ) ( ) (1 1 1) ( ) ( )[ ]k
P P N N S S
kk
E
k
W W
k
Ea a a a a b Jakob:
( ) ( ) ( ) ( )1 1 1 )1([ ]k k k k
P P N N S S E E W W
ka a a a a b G-S:
Scanning （扫描）in E-W direction
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6.2.3 Alternative direction iteration－ADI
1. Basic idea
First direct solution for each row(行)（or column 列），then direct solution for each column（or row)；The combination of the two updating of the entire domain consists of one cycle iteration:
Alternative direction iteration
(ADI) vs. alternative
direction implicit (ADI)：
It can be shown that: one-time step forward of
transient problem is equivalent to one cycle iteration
for steady problem (see appendix).
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ABEqs. generated on structured grid system can be
solved by ADI.
2. ADI-line iteration is widely adopted in the numerical
solution of flow and heat transfer problem.
ADI-iteration of solving multi-dimensional steady
problem for one iteration (ADI-iteration) is very similar
to the ADI-implicit of solving multidimensional unsteady
problem for one time step (ADI-implicit).
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6.3.1 Sufficient condition for iteration
convergence of Jakob and G-S iteration
6.3.2 Analysis of factors influencing iterationconvergence speed
6.3 Convergence Conditions and AccelerationMethods for Solving Linear ABEqs.
6.2.3 Methods for accelerating transferringboundary condition influence intosolution domain
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6.3 Convergence Conditions and AccelerationMethods for Solving Linear ABEqs.
6.3.1 Sufficient condition for iteration convergence of Jakob and G-S iteration
Coefficient matrix is non-reducible (不可约), and is
diagonally predominant (对角占优）:
1. Sufficient condition－Scarborough criterion
2. Analysis of coefficients of discretized diffusion-
convection equation by the recommended method
1nb
P
a
a
1 for all equationsat least for one equations1
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1) Matrix is non-reducible－If matrix is reducible then
the set (集合） of coefficients subscript (矩阵下标) ,W ,
can be divided into two non-empty (非空) sub-sets, R and
S ，W＝R＋S，and for any element from R and S, say k
and l respectively，we must always have： ；If
such condition does not exist, then the matrix is called
non-reducible (不可约)
, 0k la
Analysis：Coefficient of discretized equation represents
the influence of neighboring nodes. For nodes in elliptic
region any one must has its effects on its neighbors; If
matrix is reducible it implies that the computational domain
can be divided into two regions which do not affect each
other---totally impossible .
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Non-reducible matrix is determined by the physical fact that neighboring parts in flow and heat transfer are affected each other.
2) Diagonally predominant－Coefficients constructed
in the present course must satisfy this condition:
(1) Transient and fully implicit scheme
0
P nb P Pa a a S V 0, 0, 0P Pa S , P nba a
(2) Steady problem with non-constant source term
0PS , P nba a
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(3) Steady problem without source term
For inner grids:P nba a
At least one node in the boundary can WT
P P E E W W N N S Sa T a T a T a T a T b
is solved, it becomes：
0 ( )P P E E N N S WS Wa T a T a T a T b a T
Hence here： 0E NP nb Sa aa aa 2) For 3rd kind boundary condition，additional source term helps
0PS , = )P nb P nba a S a -（-
P nba a1)Assuming that Tw is known，then when the eq.
be found to satisfy ：
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It is impossible that all boundary nodes are of 2nd
type, at least one node is of 1st or 3rd type. Otherwise
there is no definite solution!
6.3.2 Analysis of factors influencing iterationconvergence speed
1. Transferring effects of B.C. into domain---View P.1
The steady state heat conduction with constant
properties are governed by Laplace equation,
for which a uniform field satisfies. However, it is
not the solution because B.C. is not satisfied.
2 0
Thus numerical methods recommended by the present course must satisfy this sufficient condition
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Thus the transferring speed for the effects of boundary
condition must affect iteration convergence speed.
2. Satisfaction of conservation condition---View P.2
For a problem with 1st kind boundary condition, itis possible to incorporate all the known boundary values into the initial field, but such an initial field does not satisfy conservation condition. Thus techniques which is in favor of satisfying conservation condition can accelerate convergence speed；
3. Attenuation (衰减）of error vector---View P.3
The error vector is attenuated during iteration. Error
vector is composed of components of different frequency.
Techniques which can uniformly attenuate different
components must can accelerate convergence speed.
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Taking the numerical error of each node as a
component of a vector, then all the error components
consist a vector, called error vector.
Error curve
The error curve can be decomposed by a number
of sine/cosine components with different frequencies.
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4. Increasing percentage of direct solution---View P.4
Direct solution is the most strong technique that both
conservation and boundary condition can be satisfied.
Thus appropriately increasing direct solution proportion
is in favor of accelerating convergence speed.
6.3.3 Techniques for accelerating transferringB.C. effects
Jakob iteration：In each
cycle the effect of B.P. can
transfer into inner region by
one space step. Very low
convergence speed.
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G-S iteration：The
effects of the iteration
starting boundary are
transferred into the entire
domain; Convergence
speed is accelerated.
Line iteration：The
effects of iteration starting
boundary and the related
two end boundaries are all
transferred into the entire
domain; convergence
speed is further accelerated.
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ADI line iteration：In every
cycle iteration effects of all the
boundaries are transferred into
the entire domain. The fastest
convergence speed.
ADI line iter.>Line iter.>G-S iter.>Jakob iter.
Jakob iteration has the slowest convergence speed.
That is the change between two successive iterations is
the smallest; This feature is in favor of iteration
convergence for highly non-linear problems when iteration
cycle number is specified. In the SIMPLEST algorithm,
Jakob iteration is used for the convective part of ABEqs.
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6.4.1 Necessity for block correction technique
6.4.2 Basic idea of block correction
6.4 Block Correction Method –Promoting
Satisfaction of Conservation
6.4.3 Single block correction and the boundary condition
6.4.4 Remarks of application of B.C. Technique
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6.4 Block Correction Method –Promoting Satisfactionof Conservation
6.4.1 Necessity for block correction techniqueFor 2-D steady heat conduction shown below when
ADI is used to solve the ABEqs. convergence speed is very low：EW boundaries have the strongest effect because of 1st kind boundary, but the influencing coefficient is small ；N-S boundary is adiabatic, no definite information can offer, but has larger coefficient－Thus to accelerate convergence of solving ABEqs., a special method is needed
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6.4.2 Basic idea of block correction
Physically, iteration is a process for satisfying
conservation condition；In one cycle of iteration, a
correction, , is added to previous solution, ，which does not satisfy conservation condition, such
that ( ＋ ) can satisfy conservation condition
better. The process of solving ABEqs. of is the
process of getting the solution of .
*
'
* '
'
'
j
For 2-D problem, corrections are also of 2-D;
In order that only 1-D corrections are solved, corrections
are somewhat averaged for one block, denoted by
or , and it is required that ( ) or ( ) satisfies the conservation condition.
'*
,i j i
'*
,i j j
'
i
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6.4.3 Single block correction and the boundary condition
* * *
, 1, 1,
*
, 1
*
1 1
, 1
( ) ( ) ( )
( )( )
( )( )
( ,.... 2)
i j i j i j
j j j
i j i
j
i j i
j
i i i
j
AP AIP AIM
AJM
AJP CON
i IST L
IST-starting subscript in X-direction；L2-last but one.
1.Equation for correction：'*
,( )i j i It is required that： satisfy following eq.
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Rewrite into ABEqs. of ：' ' '
1 1, ,
i i i
' ' '
1 1( ) ( ) ( ) , ,.... 2
i i iBL BLP BLM BLC i IST L
where
2
2
( ) ( ) ( )M
j JST j M i JST
BL AP AJP AJM
2
( )M
j JST
BLP AIP
；2
( )M
j JST
BLM AIM
2 22* *
, 1 , 1
2 2 2* * *
1, 1, ,
( ) ( )
( ) ( ) ( )
M
i j i j
j JST
M M M
i j i j i j
j JST j JST j
M M
j JST j J
JST
ST
BLC CON AJP AJM
AIP AIM AP
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2
2
( ) ( ) ( )M
j JST j M i JST
BL AP AJP AJM
ASTM is adopted to deal with 2nd and
3rd kind boundary condition，this is
equivalent to that all boundaries are
of 1st kind, and the correction for
boundary nodes is zero；Thus when
summation is conducted in y-direction
the 1st term and the last term corrections
are zero. Hence, for AJM term JST is not
needed，and for AJP term M2 is not
needed.
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6.4.4 Remarks of application of B.C. technique
1.BCT is not an independent solution method. It should
be combined with some other method, such as ADI；
2. For further accelerating convergence ADI block
correction may be used.；
3. For variables of physically larger than
zero values the BCT may not be used
(such as turbulent kinetic energy,
component of a mixed gas). Because BCT
adds or subtracts a constant correction
within the entire block, which may lead to
negative values.
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6.5.1 Error vector is attenuated(衰减) in the
iteration process of solving ABEqs.
6.5.2 Basic idea and key issue of multigridtechnique
6.5 Multigrid Techniques –Promoting
Simultaneous Attenuation of Different
Wave-length Components
6.5.3 Transferring solutions between differentgrid systems
6.5.4 Cycling patterns between different gridsystems
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6.5 Multigrid Techniques –Promoting Simultaneous Attenuation of Different Wave-length Components
6.5.1 Error vector is attenuated in the iterationprocess of solving ABEqs
Taking 1-D steady heat conduction problem as
an example to analyze how error vector is attenuated:
1. How error vector is attenuated during iteration?
2
20
d Tf x
dx
Discretizing it at a uniform grid system, yielding:
2
1 12 ( )i i i iT T T x f
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( ) ( ) ( 1) 2
1 12 ( )k k k
i i i iT T T x f
In the kth cycle iteration error vector is denoted by( )k
( )k
iand its component is denoted by
( ) ( )k k
i i iT T
Substituting this expression to the above equation we
can get following variation of error with iteration
, then we have:
Adopting G-S iteration method from left to right：
( ) ( ) ( 1) 2
1 12 ( )k k k
i i i iT T T x f
( ) ( )
-1 -1 -1-k k
i i iT T ( ) ( )-k k
i i iT T
( ) ( )
1 +1 +1-k k
i i iT T
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2. Analysis of attenuation of harmonic components
( )
( 1) 2
I
I
k e
k e
Amplifying factor
(增长因子)
It will be shown later that ( )k
i can be expressed as：
is the phase angle，by substituting this expression to the
above eq.，yielding
( ) Iik e where is the amplitude (振幅）and ( )k
2
1 12 ( )i i i iT T T x f Since
( ) ( ) ( 1)
1 12 0k k k
i i i
Then we have:
This equation presents the variation of error with iteration.
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Analyzing amplifying factor for different phase angles：
, cos sin
2 cos sin
I
I
Ite.5 times
/ 2,
cos sin2 2
2 cos sin2 2
I
I
Ite.5 times
/10,
cos sin0.9510 0.3090 110 10
,2 (0.9510 0.3090 ) 1.094
2 cos sin10 10
II
II
Ite.5 times
0
0
0
0
5 30.333 4.09 10
50.447 0.0178
50.914 0.658
1 1,
2 1 3
2
1 1,
52 1
2xk x x
From above calculation phase angle can be an
indicator for short/long wave components.
/ 2
Generally for components with phase angle within
following range it is regarded as short wave ones:
where is the wave length. At a fixed space step,
short wave has a larger phase angle, and is attenuated
(衰减)very fast; while long wave component has
small phase angle and attenuated very slowly.
can be expressed by：
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In such a way by amplifying space step (放大空间步长）several times during iteration all the error
components may be quite uniformly attenuated and the
entire ABEqs. may be converged much faster than
iteration just at a single grid system.
This is the major concept of multigrid technique
for solving ABEqs.
This phase angle is dependent on space step length
. If after several iterations the
length step is amplified then originally long wave
component may behave as a short wave and can be
attenuated very fast at that grid system.
2 /xk x x
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6.5.2 Major idea and key issue of multigridtechnique
1. Major idea－Solving ABEqs. is conducted atseveral grid systems with different space step length such that error components with different frequencies can be attenuated simultaneously.
2. Key issues－(1) How to transfer solutions at different grid systems?
(2) How to cycle (轮转）the solutions between several
grid systems?
6.5.3 Transferring solutions between two girdsystems
Basic concept： solution transferred between different
grid system － is the one of the finest grid.
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Taking two grid systems, one coarse (k-1) and one fine (k), as an example to show the transferring of solutions.
( 1) ( 1) ( ) ( )( 1) ( )1( )k k k kk kk
kA b I b A
Residual of fine grid
Operator for transferringform kth grid to (k-1)thgrid
Source term at (k-1)thgrid determined from solution of kth grid
Solution at (k-1)th grid
Matrix at
( k-1)th grid
determined
from
solution of
kth grid.
1.From fine grid to coarse grid
2. Transferring from coarse grid to fine grid
( ) ( 1) ( )1
1( )k k k kk k
k krev old oldI I
Solution of kth grid
expressed at (k-1)th
grid
New solution of fine
grid obtained at
( k-1)th grid
Original solution at
fine grid
Operator for transferring
correction part of solution
at (k-1) th grid to kth grid
Revised
solution at
fine grid
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3. Restriction and prolongation operators
Direct injection(直接注入)
Nearby average(就近平均),Linear interpolation
Near average
fine course
For node 4
direct injection
（From fine to course）
1) Restriction
operator(限定算子）
1k
kI
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Linear interpolation
Quadratic interpolation
(二次插值)
2) Prologation
operator
(延拓算子) 1
k
kI
（From course
to fine）
FineCourse
Node 4-
Direct
injection
Linear
interpolation
between nodes
3，4Quadratic
interpolation
Direct injection
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V-cycle W -cycle
Number in the circle shows times of iteration. Black
symbol represents converged solution. FMG cycle is
widely adopted in fluid flow and heat transfer problems.
6.5.4 Cycling method between several grids
Three cycling patterns：
FMG-cycle
同舟共济渡彼岸!People in the same boat help each other to cross to the other bank, where….
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